Use the graph to determine the domain of \( y=-2 \cot x \). A. \( \left\{x \left\lvert\, x \neq \frac{k \pi}{2}\right., k\right. \) is an odd integer \( \} \) B. \( \left\{x \left\lvert\, x \neq \frac{k \pi}{4}\right., k\right. \) is an odd integer \( \} \) C. All real numbers D. \( \{x \mid x \neq k \pi, k \) is an integer \( \} \)

To determine the domain of the function \( y = -2 \cot x \), we need to consider the properties of the cotangent function. The cotangent function, \( \cot x \), is defined as: \[ \cot x = \frac{\cos x}{\sin x} \] The cotangent function is undefined wherever the sine function is equal to zero, since division by zero is undefined. The sine function is zero at integer multiples of \( \pi \): \[ \sin x = 0 \quad \Rightarrow \quad x = k\pi, \quad k \in \mathbb{Z} \] Thus, \( \cot x \) is undefined at these points, which means that \( y = -2 \cot x \) is also undefined at these points. Now, we need to find the correct option that describes the domain of \( y = -2 \cot x \): - **Option A**: \( \left\{x \left\lvert\, x \neq \frac{k \pi}{2}\right., k\right. \) is an odd integer \( \} \) - This is incorrect because \( \frac{k \pi}{2} \) does not cover all points where \( \cot x \) is undefined. - **Option B**: \( \left\{x \left\lvert\, x \neq \frac{k \pi}{4}\right., k\right. \) is an odd integer \( \} \) - This is incorrect because \( \frac{k \pi}{4} \) does not correspond to the points where \( \cot x \) is undefined. - **Option C**: All real numbers - This is incorrect because there are points where \( \cot x \) is undefined. - **Option D**: \( \{x \mid x \neq k \pi, k \text{ is an integer} \} \) - This is correct because it accurately describes the points where \( \cot x \) is undefined. Therefore, the correct answer is: **D. \( \{x \mid x \neq k \pi, k \text{ is an integer} \} \)**